# Why work is defined as a scalar?

• Tahmeed
In summary: The vector equation for work is:\frac{\partial}{\partial x} dx = \frac{F_x(x)}{r}In summary, work is a scalar quantity that is used to measure the amount of force needed to achieve a goal. It was convenient to give it a name, F.s, and there is no real reason for the definition other than convenience.
Tahmeed
we know work is F.s which is a scalar quantity. but why it is defined this way? work actually has something to do with direction. Negative work means reduction in velocity and velocity has direction. What was the problem of defining work as vector that lies in the direction of displacement??

Tahmeed said:
work actually has something to do with direction.
It does not.
Tahmeed said:
Negative work means reduction in velocity and velocity has direction.
It does not have to be a reduction in velocity, it can be everything. Also, you can change the direction of velocity without altering speed. Do you want to have a separate work quantity for every possible direction where deceleration can occur?
Tahmeed said:
What was the problem of defining work as vector that lies in the direction of displacement??
It does not make sense because work does not care about the direction of motion of whatever object was used. It's like assigning a temperature a material property based on the material used to measure it. "It is 30 degrees mercury-celsius today" - "oh, according to my thermometer it is 30 degrees semiconductor-celsius".

Tahmeed said:
we know work is F.s which is a scalar quantity. but why it is defined this way?
The quantity F.s is a useful quantity. Because it is useful, it is used often, and it was therefore convenient to give it a name. There really isn't any reason for a definition other than usefulness and convenience.

Tahmeed said:
What was the problem of defining work as vector that lies in the direction of displacement??
You mean something like ##|\mathbf{F}|\mathbf{s}##? Where is such a quantity used? I cannot think of a single time I have ever seen that quantity used.

DEvens
Let's look at work as force acting over a distance.

First the force vector:

Fortunately it has units of work, not force, and everything makes sense.

$W[ Newton \cdot meter ] = F_x[Newton] dx[meter] + F_y[Newton] dy[meter] + F_z[Newton] dz [meter]$

Normally, you are taught that $F = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$.

i, j and k are unit vectors in the x, y, and z directions, respectively. The little dx, dy, and dz replace these.

Force is a "covector" having units of work. The vector elements have units of force.

$$w = F_x dx + F_y dy + F_z dz$$
Integrated over a spatial interval, $W_{ab} = \int_{a}^{b} w$

Tahmeed said:
we know work is F.s which is a scalar quantity. but why it is defined this way? work actually has something to do with direction. Negative work means reduction in velocity and velocity has direction. What was the problem of defining work as vector that lies in the direction of displacement??
Work is supposed to be a measure of how "hard" it is to do something. If work had been a vector in the direction of the displacement, then if you push a heavy object 5 meters in one direction and then push it back, you will have performed zero work.

Also, what if you push an object in a curved line? What would be the direction of the work vector then?

Mathematically, work is the dot product between the force and distance. Dot product creates a scalar out of two vectors. Note that if you push the side of a truck that happens to be moving forward (perpendicular to your force), then you aren't doing work on the truck. The dot product converts the two directions (your force and the trucks motion) into a single quantity, independent of direction, since you can rotate the entire coordinate system and you still do no work on the truck.

Work shares the same units as energy, and energy has no direction. In fact, in special relativity, energy and momentum form a 4-vector. You could think of the energy as the component of energy-momentum directed in the time direction, while the momentum is the three components directed in space directions.

rumborak
stedwards said:
Force is a "covector" having units of work. The vector elements have units of force.
What? That makes no sense. Force has units of mass*length/time^2. Work has units of mass*length^2/time^2.
Note that length^2 = length dot length, which is a scalar.

$W[ML^2T^{-2}] = F_i[MLT^{-2}]dx^i[L]$

$W_{ab} = \int_a^b F_i dx^i = \int_a^b W$

Re. length squared as a scalar. Interesting.

$\frac{\partial}{\partial x} dx$ is scalar and dimensionless.

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Work is a quantity that just tell you how much the force achieved its "Goal"(the goal here is the displacement from one place to another with the direction of the force ).
suppose that you are making a force on a block,but the block didn't do displacement at all.Then in this case the work is zero(you didn't achieve your goal).
If you want to multiply vectors like how you you multiply real numbers,the two vectors must be in same direction else you should project one into another(and that what scalar product does).
W=F.r, you should project F in the direction of r because it is the condition where your goal is achieved or oppositely achieved (if your force is projected in the same direction but in opposite sense with displacement).
F is the force and r is the displacement.If the block is displaced and there is zero force then work is zero,and if there is forced applied on the block but no displacement occurred then work is zero either.
SO WORK CAN'T BE A VECTOR QUANTITY.IT JUST TELL YOU HOW MUCH YOUR GOAL IS ACHIEVED. Look at the figure below.

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## 1. Why is work defined as a scalar?

Work is defined as a scalar because it only has magnitude and no direction. It is a scalar quantity because it is a measurement of the amount of energy transferred when a force is applied over a certain distance.

## 2. How is work calculated as a scalar quantity?

Work is calculated as a scalar quantity by multiplying the force applied on an object by the distance it moves in the direction of the force. The formula for work is W = F * d, where W is work, F is force, and d is distance.

## 3. Can work ever be negative if it is a scalar quantity?

Yes, work can be negative even though it is a scalar quantity. This occurs when the force applied is in the opposite direction of the motion, resulting in negative work being done.

## 4. Why is work considered a fundamental concept in physics?

Work is considered a fundamental concept in physics because it is a crucial part of understanding energy and motion. It helps us quantify the amount of energy transferred and the effect of forces on an object.

## 5. How is work related to power?

Work and power are closely related concepts. Work is the amount of energy transferred while power is the rate at which work is done. The formula for power is P = W/t, where P is power, W is work, and t is time.

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